TITLE: Problems from the Kourovka Notebook that undergraduate students can fully appreciate
QUESTION [20 upvotes]: The Kourovka Notebook is a collection of open problems in Group
  Theory. 

My question is: could you point out some (a "big-list" of) problems [by referencing them] presented in this book that are, in principle, accessible to undegraduate students: i.e., problems that refer to (and possibly might be solved by applying) definitions, concepts, and theorems that are presented in a book like Herstein's Topics in Algebra (and then, by extension, in an abstract algebra course for undergraduates).
The aim of this question is to allow undergraduate students to have a better understanding of current research in algebra by letting them see concretely open problems that can be easily related to known concepts.

REPLY [8 votes]: Problem 15.99 from the 15th edition (2002):

Let $f(n)$ be the number of isomorphism classes of finite groups of order $n$. Is it true that the equation $f(n) = k$ has a solution for any positive integer $k$? The
  answer is affirmative for all $k \le 1000$ [G. M.Wei, Southeast Asian Bull. Math., 22,
  no. 1 (1998), 93–102]. (W. J. Shi)